CoA. Note No. 132
T H E C O L L E G E O F A E R O N A U T I C S
C R A N F I E L D
GRINDING THEORY
by
M . J . Hillier
TECHNISCHE HOGESCHOOL DELFT
YUECTUIGBOUWKUNOeBIBUOTHEtK
T H E C O L L E G E O F A E R O N A U T I C S C i ; A N F I E L D G r i n d i n g T h e o r y b y -M. J . H i l l i e r SUMMARY T h i s r e p o r t p r e s e n t s a r e v i e w and e x t e n s i o n of c u r r e n t grinding t h e o r y . G r i n d i n g o p e r a t i o n s a r e c l a s s i f i e d on the b a s i s of (a) the cutting action of the g r i t , (b) the g e o m e t r y of wheel and w o r k , (c) the type of chip p r o d u c e d , (d) the e x i s t e n c e o r o t h e r w i s e of a continuous l a t e r a l t r a v e r s e of the w h e e l .
T h e t h e o r y i s developed u s i n g the above method of c l a s s i f i c a t i o n . R e c e n t e x p e r i m e n t a l work by P u r c e l l ' ^ ^ ' on the w e a r of the wheel in s u r f a c e g r i n d i n g i s r e i n t e r p r e t e d in the light of the t h e o r y . It i s shown t o be c o n s i s t e n t with the p o s t u l a t e of a c o n s t a n t t a s k p e r g r i t . A method of c a l c u l a t i o n i s s u g g e s t e d by which a s a t i s f a c t o r y s u r f a c e g r i n d i n g technique m a y be applied to the f o r m g r i n d i n g of g e a r s , w h e r e the t a s k p e r g r i t i s not independent of the i m p o s e d g r i n d i n g c o n d i t i o n s .
CONTENTS Page Summary Notation 1. 2 . 3 . 4 . 5 . 6, 7. 8. 9 . 10. 1 1 . 12. 1 3 . 1 4 . 1 5 . 16.
IT.
18. 19. 2 0 . 2 1 . 22. 2 3 . IntroductionThe Grinding Wheel
Grinding as a Cutting P r o c e s s
Grinding as a Milling P r o c e s s
Determination of Grit Surface Density
Effect of Dressing Technique on Grit Density
Classification of Grinding Operations
Calculation of Chip Thickness - Plunge Grinding
Calculation of Chip Thickness - Surface Grinding
Peripheral Side Wheel Grinding, Case (f)
Cylinder Grinding
Form Grinding
Form Grinding of Gears
Chip Classification
Grinding at high work speeds
Specific Energy
Grit F o r c e s
Cutting properties of a wheel
Intrinsic Hardness
Wheel Loading
Relative Hardness
Wheel Wear
Wear Profile in Surface Grinding
1 2 3 5 6 6 7 8 9 12 14 15 16 17 19 21 23 24 24 25 26 26 27
25. Application of Grinding Theory 35
26. Conclusions 37
27. Acknowledgements 37
28. References 38
NOTATION
b width of statistical mean chip
C number of active grits per unit surface area
D wheel diameter
D work diameter w
d nominal wheel depth of cut
d effective wheel depth of cut e
f c r o s s feed/rev of dressing tool
c r o s s feed/traverse in reciprocating surface grinding
feed/grit in cylindrical grinding
k number of active grits per unit length
t undeformed chip length m mean spacing between grits
N angular speed of wheel (rev/unit time)
n number of fresh active grits per unit length of path of dressing tool
p pitch of grits relative to workpiece
q distance moved by work during time of contact of a grit
r ratio of chip width to thickness
B feed rate per unit time
t , , maximum undefornaed chip thickness M
T time taken for work to move a distance p
T time of contact of grit with work
V surface speed of wheel
V work speed
w' effective width of wheel
similar manner. The theory of grinding is concerned, therefore, with the geometry of chip formation. This is of practical interest since (a) the maximum force on a grit is a function of the chip thickness, (b) the total volume of a chip nnust be accommodated by the
pores in the bond material holding the g r i t s . The force on a grit governs the tendency to lose g r i t s , and hence the rate of wheel wear. Similarly, if a chip cannot be accommodated by the available voids in the wheel it will be dragged across the ground surface and so damage it.
In a theoretical analysis of cylindrical grinding Guest distinguished between (a) the depth of cut as measured by the total thickness of metal removed from the surface per pass (the infeed, wheel or radial depth of cut) and, (b) the thickness of a chip as seen by a grit (the chip or grit depth of cut). As in milling, the grit depth of cut, i . e . the chip thickness, is limited by the number of cutting elements in simultaneous contact with the workpiece in the plane of rotation. Guest further assumed that each chip was of triangular cross section; the chip cross sectional area, and therefore the force per grit, then varies as the square of the chip thickness.
(2)
In a similar analysis Alden obtained an expression for the maximum chip thickness which reduces to Guest's formula when the length of the chip is small compared with both wheel and work diameter. The force on a grit was assumied to vary directly as the chip thickness.
(3)
Hutchinson differentiated between upcut and downcut grinding and calculated the length of the chip and the chip thickness. The latter was calculated, however, assuming that the average grit spacing was equal to the length of the chip. This implied that only one grit was in contact with the work at any instant. If this assumption is corrected Guest's formula is obtained. Hutchinson also considered problems of work heating, chip formation and wheel dressing.
More recent theories of grinding are due to, or derive from, the work of Shaw and his associates • • • > • _ These theories are concerned with the definition of the mean grit, the mean grit spacing, the mean chip and the width of a statistical mean chip. Grinding theory has also been extended to cover extreme conditions peculiar to internal grinding, very light depths of cut and high work s p e e d s ' " ' . The reference cited suggests a classification of types of chip based on the geometry and shape of the chip. Finally, the theory has been used by Shaw'^^' to indicate how grinding burn, residual s t r e s s e s due to grinding, and surface cracking may be minimised.
(12)
Experimental work by Purcell on the wear of the wheel in surface grinding has shown that the profile of the wheel periphery tends to take up a characteristic profile, depending upon the imposed grinding conditions, so as to impose a constant task per grit.
The present report attempts to classify grinding processes and presents a theoretical analysis based on this classification. The suggested classification is based on considerations of: the cutting action of the grit, the geometrical configuration of the wheel and work, and the type of chip produced. It will be shown how the postulate of a constant task per grit may be incorporated in current theory. A method of applying satisfactory surface grinding techniques to the form grinding of gears is outlined. Finally, the classification of types of chip presented in reference (8) is extended and the definition made more precise; and the theory of grinding at high work speeds is extended to cover a practical case not considered in the l i t e r a t u r e .
2
-2. The Grinding Wheel
The grinding wheel is a disc or cylinder composed of abrasive particles embedded in a porous matrix. The abrasive particles, known as grits or grains, present sharp edges to the work. The action is a true cutting p r o c e s s , but more complex than is the case with a lathe tool due to the irregular shape of the g r i t s . Similarly the size of the grits is not uniform but is controlled within liniits. The porous matrix is composed of the bond material and pores or voids. The bond serves to provide tool posts (bond posts) to hold the abrasive g r i t s . The pores are spaces not occupied by either grits or bond material and they provide accommodation for the chips removed by the cut. Ideally each cutting grit is acconapanied by a void sufficient to c a r r y the cut chip. This is then washed away by the cutting fluid as the grit clears the workpiece.
The inherent properties of a wheel depend on the type of abrasive, the size and
distribution of the g r i t s , the amount and type of bond material and the volume of the pores relative to that of grits and bond. The inherent wear properties of a wheel are determined by the resistance to abrasion of the grits, their strength, and the strength of the bond posts. The resistance to abrasion is a function of the type of abrasive and may be modified by the nature of the material of the workpiece and the chip-grit Interface temperature. The strength of a grit is the resistance offered to fracture during cutting. It is a function of the type of abrasive and of the grit size. The strength of the bond is the resistance offered to breakdown of the bond posts and consequent loss of g r i t s . It is a function of the type and relative amount of bond material.
Typical abrasives a r e :
aluminium oxide (alundum), denoted by the letter - A silicon carbide (carborundum) - S
diamond - D
The term grit size relates to the fineness of the mesh used in grading the abrasive g r i t s . the larger the 'size' the finer the grit. The range of grit sizes is as follows:
coarse 6 - 2 4 medium 30 - 60
fine 70 - 600
(9) The grit size (S) is related to the grit diameter (g in.) by the approximate formula
S.g = 0.6
Thus a grit size of 30 corresponds to a grit diameter of 0.02 in,
The wheel grade is a measure of the inherent strength or hardness of the wheel, i . e . the resistance to breakdown and wear. Since under correct cutting conditions wheel wear takes place by abrasion and bond post fracture the grade or hardness is a function of the relative amount of bond post material. The resistance to abrasion of the grits is not usually considered in the classification of a wheel. The wheel grades a r e :
very soft soft medium hard very hard C H L P T - G - K - O - S
- z
The term structure refers to the porosity of the wheel. Alternatively it may refer to the grit spacing. The range of porosity is from dense (O) to porous (15).
The following bond materials are generally used:
vitreous or vitrified bond (baked clay) - V silicate - S
resinoid - B rubber - R shellac - S oxychloride - O
Using the above notation a typical wheel might be classified as follows:
A 36 L 8 V
This r e p r e s e n t s a wheel having an aluminium oxide abrasive (A), a grit size of 36 mesh, a medium strength bond, i . e . a hardness or wheel grade L, a fairly high grit spacing or porosity (8), and a vitrified bond (V).
3. Grinding as a Cutting P r o c e s s
Grinding is a true cutting process. Microscopic examination of a taper section of a ground surface reveals minute Vee shaped s c r a t c h e s . Microscopic examination of grinding swarf reveals very minute chips formed by the cutting action of individual g r i t s . The cutting process at an individual grit takes place however at a higher cutting speed and with a
sinaller depth of cut as compared with cutting with a single point tool. Typical values are a s follows:
Cutting Grinding cutting speed (ft/min) 40 - 400 3,000 - 6.500
chip thickness (in.) 0.005 - 0.020 0.00003 - 0.00010
Later it will be found convenient to consider grinding as a cutting process geometrically akin to milling. The following features are peculiar to grinding however: The size and shape of the grits varies in a random manner within the limits of a specified grit size. The grits are set in the wheel in a random manner, and not all grits visible on the surface of the wheel take part in cutting. Those that do are termed active grits. Finally, the deeper the cut the greater the number of active grits that may be expected to be in contact with the work surface. Current grinding theory attempts to account for these conditions excepting the last. It is assumed that the number of active cutting grits per unit a r e a of the wheel surface is independent of the depth of cut.
4
-Cutting action of a grit
In orthogonal cutting with a single point tool the chip width is uniform and equal to the width of the tool. Consider, however, orthogonal cutting with the Vee shaped tool shown. Fig. 1. A simple check, using plasticene as a workpiece, will show that if a chip is formed on the flat face of the tool its shape will be roughly as indicated. The ratio r of chip breadth b' to depth of cut t is a constant for a given tool; thus
b'
— - V
A real grinding grit, even when sharp, does not have such a simple shape. However observation of the Vee shaped profile of the scratches formed onthe surface of a ground workpiece suggests that the above relation holds approximately, at least for surface grinding at the periphery of the wheel. Note that for this type of chip the cross sectional area varies as the square of the depth of cut.
The grit depth of cut is the undeformed chip thickness seen by an individual grit. According to the type of grinding involved the chip thickness may vary along the length of the chip. The chip width varies accordingly. Moreover, a grit might cut into a
previously uncut surface, or its path might overlap the cut made by a preceding grit. A precise specification of actual chip cross section is therefore difficult. On the other hand a knowledge of the actual variation in chip cross section along the chip, or from chip to chip, is not usually necessary. Since the uncut surface of the workpiece may be formed from a sufficient number of undeformed chips of uniform cross section so as to leave no voids it is convenient to define a statistical mean chip of uniform width which is assumed to be the same for all chips. The cross sectional area of such a mean chip varies as the chip thickness and not as the square of the thickness.
From the point of view of the cutting action of the grit it is convenient to classify grinding processes as:
1. (a) surface grinding (b) plunge grinding
2. (a) peripheral grinding (b) side-wheel grinding
In surface grinding the plane of naotion of the workpiece is parallel to the plane of motion of the cutting g r i t s . In plunge grinding, on the other hand, the motion of the work is normal to the direction of motion of the active wheel surface,
Peripheral grinding refers to a cutting process which takes place at the periphery of the wheel. Side-wheel grinding refers to a process in which the active grits are at the side face of the wheel. In practice, due to non-uniform wheel wear it may happen that the cutting action nnay be intermediate between these two extremes,
In peripheral surface grinding the width b of the statistical mean chip is defined in t e r m s of the mean chip thickness. In surface grinding, as in cylindrical or slab nnilling, the undeformed chip thickness varies from zero at the start of the cut to a maximum, t , near the end of the cut, the mean chip thickness is therefore i - t . . to a very good
approximation, and hence
In surface grinding at the side face of the wheel. Fig. 2(b), the width of the chip is equal to the depth of cut, or infeed per pass, of the wheel and independent of chip thickness.
For plunge grinding at the periphery of the wheel. Fig.2(a), the chip thickness may be shown to be constant over the length of the chip. The width of the mean chip is defined as
b = r . t
where the chip thickness t is equal to the Infeed per grit. Thus, if v is the velocity of the work normal to the periphery of the wheel and V the surface speed of the wheel,
t = ^ . m
When plunge grinding at the side face of a wheel the width of the mean chip is again a constant, equal to the infeed of the work per grit. Thus
b = ^ . m
where v Is the speed of the work normal to the side face of the wheel.
4. Grinding as a Milling P r o c e s s
The geometry of grinding will be treated in each case as for an analogous milling p r o c e s s . Before doing so, however, it is necessary to define the nunnber of active grits or teeth in a plane of rotation. In practice it is usually possible to determine by experiment the number of active grits C per unit area of the wheel surface. The number of active grits k per unit length of the surface, measured in a plane of rotation, is the number of grits contained in the strip AB, Fig. 3 , cut out of unit area of the cutting surface of the wheel.
In peripheral grinding the width of the strip is equal to the width of the mean chip. In side wheel grinding the strip AB becomes a curved strip of unit length. Its width is measured in a plane of rotation and it is logical therefore to take the defining width as equal to the miean chip thickness. It is assumed that the number surface density C of active grits is uniform. The mean grit or tooth is now defined as a tooth having a width equal to the width of the defining strip AB. The mean distance between successive grits is of course 1/k.
In peripheral grinding the width of the strip AB is equal to b . Then
k = 1/m =» C. (area of strip) =• C . b . l
» C.b
For surface grinding b = 2 • r . t . , M
Hence 2
6
-S i m i l a r l y , for plunge g r i n d i n g b = r t and m =
C . r . t
In s i d e - w h e e l s u r f a c e g r i n d i n g the width of the defining s t r i p i s equal to the m e a n chip t h i c k n e s s 2 ' t „ . T h u s
c . t , „ M S i m i l a r l y , for s i d e - w h e e l plunge g r i n d i n g ,
1
Note that in g e n e r a l the value of mi s o defined i s l e s s for p e r i p h e r a l g r i n d i n g than for s i d e -wheel g r i n d i n g by a f a c t o r r .
T h e i d e a l wheel i s now defined a s a m i l l i n g c u t t e r having t e e t h u n i f o r m l y s p a c e d a d i s t a n c e m at the cutting r a d i u s , e a c h of which c u t s out the s t a t i s t i c a l m e a n c h i p , 5, D e t e r m i n a t i o n of G r i t S u r f a c e D e n s i t y
The n u m b e r of active g r i t s p e r unit a r e a (C) m a y be found e x p e r i m e n t a l l y a s follows: A thin l a y e r of soot i s s p r e a d evenly on a ground m e t a l s u r f a c e and the w h e e l allowed t o r o l l o v e r the s u r f a c e under i t s own weight. If the wheel be now allowed to r o l l o v e r a flat s h e e t of white p a p e r a soot r e p l i c a of the s u r f a c e g r i t s i s obtained. The n u m b e r p e r unit a r e a m a y t h e n b e estimiated. It i s a s s u m e d that the v a l u e of C found b y t h i s m e t h o d i s a f a i r e s t i m a t e of the active g r i t s u r f a c e d e n s i t y . The effect, if any, of the depth of cut i s not allowed f o r . The value of C obtained i s slightly dependent on the load on the w h e e l , 6. Effect of D r e s s i n g T e c h n i q u e on G r i t D e n s i t y
The d r e s s i n g of a grinding w h e e l c o n s i s t s in t r a v e r s i n g a diamond point o r o t h e r tool a c r o s s the face of the r o t a t i n g w h e e l . T h e effect i s t o b r e a k out w o r n g r i t s and loaded p o r e s and p r e s e n t a s e t of f r e s h , s h a r p , g r i t s . The s u r f a c e d e n s i t y C, and h e n c e the n u m b e r of t e e t h p r e s e n t e d to the w o r k , i s a function of the speed of t r a v e r s e of the d r e s s i n g t o o l . T h i s nnay be shown a s follows:
A s s u m e that the d r e s s i n g tool p r o d u c e s n f r e s h g r i t s p e r unit length of i t s path r e l a t i v e to the wheel f a c e . Let the feed of the tool a c r o s s the wheel for e a c h r e v o l u t i o n be f. T h e developed length of the w h e e l c i r c u m f e r e n c e i s TT.D, F i g . 4 . It i s convenient to c o n s i d e r a width b of the wheel face equal to the m e a n chip width in the s u b s e q u e n t g r i n d i n g o p e r a t i o n . In the figure AB i s a path of the tool r e l a t i v e t o the wheel. It m a k e s an angle 6 with the end face of the w h e e l , w h e r e
f tan e TT.D
and e = helix angle of the path. T h e length of such a path i s
AB = f sin e if f i s s m a l l c o m p a r e d with D.
The number of fresh grits produced on the path AB = n (AB)
- n-f sin e
The average number of such complete paths (including the sum of incomplete paths) is b
f
Hence, the number of grits in the area considered is K = "-^ ^
sin 6 • f
and, since sin Ö rCb tan 6 = f/ir.D if 0 is small, the number of grits per unit length of circumference is 7t. n. D f f / j T . D n (f/b)
The number of 'teeth' on the wheel face therefore depends strongly on the ratio of the feed per revolution during dressing to the mean chip thickness width in the subsequent grinding p r o c e s s . Halving the relative feed (f/b) tends to increase k by a factor of four, Similarly, since an increase in the depth of cut of the dressing tool tends to increase the number of fresh grits per unit length (n) it will also tend to increase k. However, if a sufficiently large numiber of passes of the dressing tool be made the total number of fresh grits will tend to a limiting value independent of the method of dressing. Finally, it should be noted that the same dressing technique may produce a different value for k (or C)
according to the nature of the subsequent grinding operation (different b),
7. Classification of Grinding Operations
The essential similarities and differences as between one grinding process and another is brought out by a proper classification of operations. In the study of the cutting action of individual grits it has already been found convenient to distinguish between surface grinding and plunge grinding, and again between peripheral and side-wheel grinding. In surface grinding the active grits move parallel to the direction of motion of the work surface at the start of the cut. In plunge grinding the grit motion tends to be normal to the work velocity vector. Peripheral and side-wheel grinding are distinguished according as the active grits are on the periphery or the side face of the wheel. In each case the chip tends to flow radially inwards towards the wheel centre, but in peripheral grinding the chip is accommodated by the voids in the bond material and in cutting on the side face of the wheel the chip tends to move into the spaces between the g r i t s . F u r t h e r , the depth of cut normal to the side face of the wheel is always limited by the size of the grits, whereas in peripheral grinding the depth of cut seen by the wheel, as distinct from the grit, is not so limited,
8
-A further classification of grinding operations is possible according to the geometric configuration of wheel and work. Fig. 5 Illustrates the more usual processes. Each case, plunge grinding excepted, may be further subdivided into upcut or downcut grinding. In upcut grinding the wheel surface moves in the sense opposite to that of the work, in downcut grinding the active grits move in the same sense as the work.
Peripheral surface grinding, case (b) Fig. 5, is roughly analogous to cylindrical, b a r r e l or slab milling. Cases (e) and (f) are termed peripheral side-wheel grinding since, although the side face of the wheel is presented to the work surface, cutting may take place at the periphery of the wheel rather than at the side face. Geometrically these processes are roughly similar to face milling. In practice wheel wear at the active corner of the wheel may modify the assumed action. When the wheel is shaped to grind a definite formed surface, as in case (j), the cutting action in a plane of rotation is essentially peripheral grinding. True side-wheel grinding is illustrated by the process of gear generation, case (h), in which the wheel is so controlled as to present the side face tangent to the surface to be ground.
Finally, peripheral surface grinding in which the wheel has a lateral feed or t r a v e r s e parallel to its axis of rotation, may be further subdivided. In reciprocating surface grinding of a flat workpiece the t r a v e r s e is intermittent and occurs at the start of each longitudinal t r a v e r s e or cut. No lateral motion takes place during the cut and the forces on the wheel a r e restricted to the plane of motion. In cylindrical surface grinding on the other hand the lateral t r a v e r s e is continuous, the wheel moves parallel to its axis during the cut and a
t r a n s v e r s e force is exerted on the wheel in the sense opposite to that of the motion. Although the lateral t r a v e r s e rate is sm^all relative to the longitudinal work speed the transverse force is not negligible and modifies the forces on the active grits.
8. Calculation of Chip Thickness - Plunge Grinding
In plunge grinding. Fig. 6, the work is fed directly into the wheel without cross feed.
In calculating the grit depth of cut, i . e . the maximum chip thickness, the following assumptions are made:
1. The rate of feed relative to the wheel is v, normal to the wheel surface, and its magnitude is small compared with the surface speed of the wheel.
2. The chips are of uniform thickness t.
3. The length of a chip is equal to the mean width of the work in the direction of grinding.
4. The length of the path of contact between the wheel and the work is greater than the distance between successive g r i t s . The chip thickness is then governed by the distance which the wheel advances between the engagement of successive g r i t s .
5. The wheel is an ideal wheel.
The path of a grit relative to the work is AD, shown exaggerated in F i g . 6 , Vhe distance travelled by the work during the time of contact of a grit with the work is
q = DC = v. T
^ c where T is the time of contact and is equal to l/V.
The thickness of a chip is equal to the distance travelled by the work in the time T taken for a grit to move a distance equal to the grit spacing m. Thus
chip thickness t = AB = CD
= V. T .T, n i
where T = - ^ Hence t = r^. m
Since the chip is of uniform thickness 1 C . r . t
1 C.t
for peripheral grinding
for side-wheel grinding Therefore the chip thickness is given by
t ^l.
V C . r . t ,
<-^(i
i.e.
. .4^^. Ï(I)
where, for side wheel grinding, r is unity.
Since the surface speed of the wheel is V, = jr.D.N, it is seen that the chip thickness is proportional to the square root of the infeed r a t e , and inversely proportional to the square root of the wheel diameter or the speed of rotation.
9, Calculation of Chip Thickness - Surface Grinding
The following analysis is valid for surface grinding at either the periphery or the side face of the wheel, cases (b) and (e). Fig. 5.
In Fig. 7 the surface speed of the wheel is V. The work surface BCD moves from left to right with speed v. It is convenient, however, to consider the motion of the wheel relative to the work. Thus in the figure O. and 0 „ are successive positions of the wheel relative to the workpiece,
The wheel depth of cut d is the height of the uncut surface above the ground surface, i . e .
d = FA^
For most practical cases it is possible to make the following assumptions:
1. The work speed (4 to 60 ft/min) is snaall compared with the surface speed (3,000 to 6,000 ft/min) of the wheel, i . e .
V « V or ^ « 1.
2. The length t of the path of contact of a grit with the work is greater than the distance m between successive grits of the ideal wheel. This implies that more than one grit is in
10
s i m u l t a n e o u s contact with the work in a given plane of r o t a t i o n . The chip t h i c k n e s s i s t h e r e f o r e governed by the d i s t a n c e p moved b y the work in the t i m e t a k e n for a g r i t to m o v e a d i s t a n c e m round the c i r c u m f e r e n c e of the w h e e l , F i g , 8.
In upgrinding a given g r i t s t a r t s to cut at A , F i g . 7, a l m o s t v e r t i c a l l y below the wheel c e n t r e O and l e a v e s the w o r k p i e c e at C. The length * of the chip f o r m e d b y the m e a n g r i t i s t h e r e f o r e e q u a l to the length of the a r c A C , i. e,
« = A^ C
= A^ A^ + A^ C
= BC + A C, s i n c e A A = B C ,
If the t i m e t a k e n for a g r i t to t r a v e l the d i s t a n c e A C i s T , the work m o v e s a d i s t a n c e q, = C B , in t h i s t i m e , w h e r e and i , e . Hence = Ag C (1 + ^ ) Now A C = i . D . e
w h e r e 0 i s the angle A O - C ( r a d i a n ) subtended at the w h e e l c e n t r e by the a r c A C , F r o m t h e g e o m e t r y of the figure
c o s 6 ° 1 " 2' fi Hence the length of a chip i s
I = 1 ( 1 + ^ ) , D . a r c [ c o s ( l - i . | ) ] (2)
F o r downcut g r i n d i n g the s a m e e x p r e s s i o n i s valid if the work speed v i s taken n e g a t i v e . T h e t i m e i n t e r v a l between the e n g a g e m e n t of s u c c e s s i v e g r i t s i s T , = m / V . T h e d i s t a n c e t r a v e l l e d by the work in t h i s tinae i s
p = v . T V = m . -^ 2 -^ BC BC « 3 = a = V. V . V V T c T c • ^ 2 A ^ C d C + BC ,
If, in Fig, 7, CB is now interpreted as the pitch p, (compare C'B Fig. 8) then the maximum chip thickness is
t = EB, approximately, since CB is small
= CB sin e, since ECB = 0
= p sin 0
= m. — . sin 6
m . - . sin f a r e cos (1 - j . -)1
Now, for surface grinding
m C . r . t M C . t in peripheral grinding in side-wheel grinding M
Hence it follows that the chip thickness is given by
M
•-I
2
C . r — , sin I arc cos (1
b
i-i']
U V f
•N C r ' V -vj
V I d d *
V
J D D» where r is unity for side-wheel grinding.(3)
For surface grinding, case (b) Fig, 5, it is usually possible to assume that the wheel depth of cut (0.001 in, say) is small compared with the wheel diameter (8 to 24 in. say),
i , e . d « D, or — « 1
It follows that the angle 0 subtended by the ar c of contact of the wheel with the work is small (about one degree). Hence making use of this assmnption
A C = FC, approximately. Fig, 7
(OgC)" - ( O ^ F ) *
ƒ
( i . D ) ' - ( i . D - d ) " = V^D.d - d»= V D.d approximately, since d is relatively small,
Hence the length of the mean chip is now
12
-Similarly, the maximum chip thickness is now t ^ = m. ^ . sin = m. — — . 0 , since 0 is small V ^ ^ ' ^ ^ 2 ^
V o^c
V V D . d V • I T D 2.m. f,.^ V \|D It follows from the definition of m that4 V / d , . v
*M "^^cTr- V J D <">
when d/D is small compared with unity. Again r is unity for side-wheel grinding. SinceV =» J T . D . N it follows that the maximumi chip thickness is proportional to
i i l l s. i
^ " ^ c • ^ v ^ d*, D " * , N " ^
F o r example to increase the chip thickness by a factor of two it is necessary to increase the wheel depth of cut by a factor of eight,
10. Peripheral Side Wheel Grinding, Case (f)
Peripheral side-wheel grinding is a cutting process roughly similar to face milling, Fig. 10a. In the Figure the workpiece moves from left to right and the grinding wheel rotates about an axis normal to the surface of the work. It is assumed that the wheel overlaps the work and that the wheel centre is offset from the centre line of the work by an amount £. The cutting action is assumed to take place at the periphery of the wheel, i . e . on AB, Fig. 10(b).
It is convenient to consider the motion of the wheel relative to the work, i . e . as if the work were at r e s t and the wheel axis moving from right to left. Then O . , 0 „ 0„ and O. are successive positions of the wheel axis relative to the work. Similarly P ,
P a r e the corresponding positions of a mean grit. When the wheel centre is at O the grit considered is at P and the grit immediately preceding it is at P where it has just started to cut.
The distance between successive g r i t s , measured on the circumference of the wheel, is m, = 1/k ' P . P . After a time T, =m/V, the grit wiU have moved from P to P , where it s t a r t s to cut. In the same time the wheel centre will have advanced a distance p relative to the work, where
P = ° 1 ° 2 P P 1 2 = v . T m . V V
T h e Initial t h i c k n e s s of the chip at the s t a r t of the cut i s t » P P '
2 2 2
=" p . c o s 0„ a p p r o x i m a t e l y .
J*
After a f u r t h e r t i m e I n t e r v a l T the g r i t s t a r t s to l e a v e the work at P . , w h e r e
c 4 T ^ ^
c V
= t i m e of contact of g r i t with the c h i p .
H e r e t i s the length of the path ( P . P_ P . ) of the g r i t . If the t a b l e speed i s s m a l l
A O 4
c o m p a r e d with t h e p e r i p h e r a l speed of the wheel (v « V) t h e n I I s a p p r o x i m a t e l y e q u a l to the a r c P P ' P ' of t h e w h e e l c i r c u m f e r e n c e . T h u s length of c h i p , I » i D (* + ^^) w h e r e A„ and ^. a r e in r a d i a n . ^2 4 At P . the m a x i m u n i chip t h i c k n e s s i s w h e r e t . =* p . cos ^. Thus *M " P* ^^°^ *2 "^ ^°^ *4^ v . m , , , . . - ^ ( c o s ^2 + c o s ^^)
F r o m the definition of the nciean g r i t
m =« C . r . t -M Hence t ^ = _ 2 ^ v _ _ (eos ^^ + c o s *^) M o r
*M U^:irv<^°«*2^'=°«V J
14 -T h e t o t a l chip v o l u m e i s
i-V*
T h e a n g l e s ^ , ^ m a y b e c a l c u l a t e d f r o m the g e o m e t r y of F i g . 10, t h u s j . w - e ÖXU ip s i n 0 St 3i . D
w - 2 . e D w + 2 . e D a n d , s i m i l a r l y 1 1 . C y l i n d e r G r i n d i n g T h e g e o m e t r y of c y l i n d r i c a l g r i n d i n g i s e s s e n t i a l l y the s a m e a s for p e r i p h e r a l g r i n d i n g of a flat s u r f a c e provided t h a t allowance i s m a d e for the c u r v a t u r e of the w o r k p i e c e . F i g s . 9a and 9b i l l u s t r a t e i n t e r n a l and e x t e r n a l g r i n d i n g r e s p e c t i v e l y . T h e w o r k d i a m e t e r I s D . T h e wheel depth of cut i s d, = A _ F ' ,w 2 Now A _ F , F i g . 9(a),is the wheel depth of cut d' in an equivalent plane s u r f a c e
g r i n d i n g o p e r a t i o n which would give the s a m e chip length A C . T h e f o r m u l a e a l r e a d y found for s u r f a c e g r i n d i n g m a y t h e r e f o r e be applied to the p r e s e n t c a s e p r o v i d e d that the wheel depth of cut i s s m a l l c o m p a r e d with the wheel d i a m e t e r and the work d i a m e t e r and t h a t the w h e e l depth of cut i s r e p l a c e d by d ' . It i s n e c e s s a r y , t h e r e f o r e , t o c a l c u l a t e d' in t e r m s of the a c t u a l depth of cut d and the c u r v a t u r e of the w o r k p i e c e .
F r o m the figure C F = i.D,sin 0 =" i . D . s i n ^ w o r s i n 0 ^ __w s i n ^ " D O F Al a 2 , 2 . d > A l s o cos 0 =» pr—T =• 1 —— " 2 2 ^ t and Since it follows t h a t c o s ^ sin^0 s i n if) D 2 w D^ 3 3 3 O F 2(d - d') O F ' D w 2 1 - c o s 0 2 . 1 - cos <t> d ' / D - d ' ^ / D ^ (d - d ' ) / D - (d - d>)^/D^ 2 2
Solving for d ' , and r e m e m b e r i n g that d' and (d - d') m a y be neglected in compairlson with the r e m a i n i n g t e r m s , g i v e s
d. - - s
w
It may be shown that a similar expression may be obtained for internal grinding if D is taken to be negative,
w " Hence the length of a chip is
w
and the maximum chip thickness is given by
t„ = j ^ _ . I i —K^ (6)
M N C. r V J D - D_>D w
where D is positive for external grinding, negative for internal grinding.
12. Form Grinding
The principle of form grinding is illustrated in Fig, 11. The surface to be ground is inclined at an angle a to the wheel axis and the periphery of the wheel is shaped correspondingly. The thickness of material to be removed in one traverse of the wheel is x, measured normal to the ground surface. The cutting action is peripheral grinding. An ideal grit, of width b, cutting in the plane of rotation of the wheel,
commences to cut at A, s t a r t s to leave the work surface at P , and finishes the cut at C. The width of the ideal chip varies as shown in the figure. The thickness of the mean chip increases along its length essentially as in surface grinding. From the figure
d = d- - b.cot a
where the total wheel depth of cut is
d = X,sec a
Numerically, since the thickness x is small, and when the angle a is not too nearly a right angle, the arc PC may be neglected by comparison with the arc AC, Similarly, d„ is usually small compared with the effective wheel diameter in the plane of rotation of the grit considered. The chip length and thickness may therefore be calculated as for surface grinding with a wheel depth of cut d , = x. sec a. Evidently the wheel depth of cut increases as the angle a is increased.
When grinding a curved surface the above analysis remains true to a first approx-imation, Angle a is now the slope of the tangent to the formed surface at the point considered. However, when a approaches a right angle the tangent to the surface tends to be normal to the wheel axis. The wheel depth of cut is no longer small compared with the wheel diameter. Equations (2) and (3) now apply. Finally, when o is very nearly a right angle the wheel depth of cut is limited by the curvature of the profile to be ground, This case will be illustrated by reference to the form grinding of gears,
16
1 3 . F o r m G r i n d i n g of G e a r s
F i g . 12(a) i l l u s t r a t e s the f o r m grinding of a g e a r tooth of involute f o r m . OA i s the r a d i u s of the b a s e c i r c l e , d i a m e t e r D, , c e n t r e O. The t h i c k n e s s of m a t e r i a l to
D
be remioved in one p a s s of a wheel i s x , m e a s u r e d nornaal to the g e a r s u r f a c e and shown e x a g g e r a t e d in the d i a g r a m . A g r i t s t a r t i n g to cut at B on the b a s e c i r c l e would l e a v e the g e a r s u r f a c e at C, s i n c e e a c h g r i t cuts in the v e r t i c a l plane of r o t a t i o n of the w h e e l . In g e n e r a l the depth of cut v a r i e s o v e r the flank of the tooth and i s g r e a t e s t on that p a r t of the flank n e a r e s t the b a s e c i r c l e . Attention will be confined to t h i s c a s e . A l s o , in p r a c t i c e the g e a r profile m a y be undercut below the b a s e c i r c l e by an amount d a s shown. The t o t a l m a x i m u m wheel depth of cut i s t h e r e f o r e d + d .
o
T h e c a l c u l a t i o n of d m a y be c a r r i e d out a s follows:
It naay be a s s u m e d that both x and d a r e s m a l l c o m p a r e d with the b a s e c i r c l e d i a n a e t e r . S i m i l a r l y the a n g l e s 0, ^ a r e s m a l l a l s o . F i g . 12(a).
T h e r a d i u s of c u r v a t u r e r of an elenaent of a r c d s of the involute i s given by
A l s o r = ^ D, 0 by the p r o p e r t y of the involute c u r v e . d s o r i. e. = r d0 o = i D, 0 b o
ds^
= i è D^ 0 d0 b o o A l s o é = r r . w h e r e N i s the n u m b e r of t e e t h on the w h e e l . ^o N F o r the t r i a n g u l a r e l e m e n t shown dx •r- = tan {lb + e) é}i ip + e d s ^ ^o o dz -r- = s e c ( i / i + 0 ) : ^ s e c (rfi + 0 ) d s ^ ^o o and d s :iii d s o (7) H e n c e , using equation (7) , 0 xi\
di°b
/ 0 ( | : + 0 )d0 J o N o o oe°
/
^o '-' <S ^ V ^'o
(8a) (8b) »E q u a t i o n s (8a),(8b) h a v e b e e n solved n u m e r i c a l l y t o give F i g . 12(b) which p l o t s d / ( i D ) a s a function of x / ( | D, ) for a r a n g e of v a l u e s of N. F o r e x a m p l e , if x = 0.001 i n . , D - = 4 i n ; the w h e e l depth of cut i s 0.05 i n .
14. Chip C l a s s i f i c a t i o n
In c e r t a i n e x t r e m e c a s e s , for e x a m p l e i n t e r n a l c y l i n d r i c a l s u r f a c e g r i n d i n g at v e r y light d e p t h s of cut and high w o r k s p e e d s , the s h a p e and type of chip f o r m e d m a y differ f r o m the m o r e u s u a l c a s e s a l r e a d y c o n s i d e r e d . A p o s s i b l e c l a s s i f i c a t i o n of chip t y p e s h a s b e e n s u g g e s t e d ^ " ' . T h e p r e s e n t r e p o r t follows the c l a s s i f i c a t i o n offered in t h e r e f e r e n c e c i t e d , e x c e p t i n g t h a t a m o r e p r e c i s e definition of t h e t y p e s of chip will be a t t e m p t e d . T h e m e t h o d of definition r e q u i r e s a different notation and r e v e a l s c e r t a i n d i f f e r e n c e s in the m e t h o d of c l a s s i f i c a t i o n , a s will now be shown.
It w i l l b e r e c a l l e d t h a t t h e t i m e of c o n t a c t of a g r i t with the w o r k i s given by * / V and t h e d i s t a n c e t r a v e l l e d by t h e w o r k in t h i s t i m e i s q, = t , —. S i m i l a r l y , the
t i m e i n t e r v a l b e t w e e n the e n g a g e m e n t of s u c c e s s i v e g r i t s i s m / V , and the c o r r e s p o n d i n g d i s t a n c e t r a v e l l e d b y the work in t h i s t i m e i s p , = m . —. T h e m e t h o d of c l a s s i f i c a t i o n i s b a s e d on the r e l a t i v e m a g n i t u d e s of the q u a n t i t i e s p and q, a s i n d i c a t e d in the
following t a b l e : o r Chip type A B C D Definition p < q p = q m > p > q p =» m > q p > m > q m m I m ^ > > > ^ < xn I 1 > V V V V " 1 1 V V 1 1
18
-In general the chip length * is a function of the depth of cut, wheel diameter and the configuration of wheel and work. It is a function also of the ratio v/V. The spacing m between successive grits is a function of wheel structure and the method of dressing. Also, from the definition of the mean grit, it is also a function of the maximum chip thickness and therefore to some extent of the grinding geometry.
The theoretical analysis of the grinding process so far considered has been restricted to the type A chip. For this chip m is less than t and there are a number of grits simultaneously in contact with the work. The maximum chip thickness is determined by the pitch per grit p, and is always less than the wheel depth of cut.
F o r the remaining types of chip only one grit is in contact with the work at any instant; or none. The chip thickness is now determined by the magnitude of the
quantity q rather than p. Hence, at the relatively low work speeds already considered (v/V « 1 ) q is small compared with t (type B or type C chip). Also q is equal to or l e s s than p. The maximum chip thickness is now given by
t = q. sin 0 , approximately,
where 0 is the angle turned through by the wheel in time T - Hence it may be shown
that ^
-
2.«.
^. J g
= 2.d. ^ since l^^U.d
Fig. 13 illustrates the formation of the chips of type A and C. The type A chip is the type I chip of the reference cited^^'. The types lU and v chips of that reference are both type C in the present classification. The type III chip is formed when a grit is always in contact with the work. The type v chip is formed when intermittent cutting occurs as in the figure. In both cases it is evident that a type C chip is formed when the wheel depth of cut is very small; the figure shows the depth of cut exaggerated relative to the wheel diameter.
The effect of very high work speeds on A and C chips leaves the geometry essentially unaltered. However it is no longer possible to assume that p and q are small compared with i, and a modification to the theory is required when calculating the chip thickness. The length of the chip is no longer calculable by an elementary formula; however, for the type V chip it is evident that the length is very nearly twice that given by the formula
i - ( 1 ^ ^ ) ^"DTd.
Finally, chips D and E occur only when the work speed is equal to or greater than the wheel speed. For most practical purposes these conditions need not be considered.
15. G r i n d i n g at high work s p e e d s
When the w o r k speed i s not s m a l l c o m p a r e d with the s u r f a c e speed of the wheel the feed p e r g r i t i s no l o n g e r s m a l l c o m p a r e d with the a r c of contact between wheel and w o r k . The following modified a n a l y s i s for t h i s c a s e ' ° ' r e f e r s to a type I c h i p , F i g . 7, (p < q, m < I). T h e m a x i m u m chip t h i c k n e s s i s M OgE - O^B
OgE - J ( 0 2 F ) ^ + (FB)^
L e t C F = X and CB = p , t h e n F B = x - p . Hence 2 2 2 d + x + p - 2 , x . p - D . d ( | . D ) 2 =• i , D , ( l - V 1 + z )- | , D [ I -(1 + i.z - .,,)]
1 2 2 2 T h e r e f o r e t = 1 + =—r r 2 , x . p - (x + p + d ) l 2 2 2 Now d m a y be n e g l e c t e d in c o m p a r i s o n with t h e r e m a i n i n g t e r n a s ; a l s o x =• ( c F ) » D . d . H e n c e , and s i n c e m . w h e r e *M d V V ^M d B = 2 . P _ ^ .^D.d D . d 2 V ^ • - ^ M ' ^ 2 . B B* < * M / ^ ^ ( t ^ / d ) 2 2 V * A/" C . r , d . V D . d (9a) (9b) E q u a t i o n (9b) for t / d h a s t h r e e r e a l r o o t s for B l e s s t h a n 3 2 / 2 7 , F o r e a c h p o s i t i v e value of B t h e r e a r e two p o s s i b l e p o s i t i v e v a l u e s of t / d , a s shown in F i g . 15.(8)
The r e f e r e n c e cited c h o o s e s the g r e a t e r of the two p o s i t i v e v a l u e s . H o w e v e r , s i n c e p < q it i s unlikely that the undeformed chip t h i c k n e s s will a p p r o a c h the wheel depth of cut for t h i s type of chip, s i n c e t h e r e a r e a n u m b e r of g r i t s in s i m u l t a n e o u s c o n t a c t . Hence it i s suggested that the s m a l l e r value of t „ / d i s the c o r r e c t o n e . The modification of t h e o r y r e q u i r e d for c y l i n d r i c a l grinding i s given in Ref. 8.
20
-2
When v/V is small B can be neglected and the following solution is obtained, v i z . , ^ = ' V T B d i . e .
t
CH X ii
M ' N J C r ' V ^JD
The case p > q (1. e. m > * , type III chip) has already been considered in section 14 for low work speeds. It was found that the chip thickness is now governed by the magnitude of q, rather than the feed per grit. Fig. 14 illustrates this case for high work speeds. The chip thickness is BE, where CB = q. It may be shown that equation (9a) remains valid for this case if p is replaced by q in that equation.
Since q =» t.v/V it follows that t /d is now given by
*M d
The 1 ength of a chip
I where A B' and 0 «1 2. e V t^ v^ / D . d V D,d v^ i s
= A^ B' (1 +
I)
= i.D(0 + 0^) - sin 0, if 0 is small r — = 2 J ^ . if d/D is small. a r c A'A £: p / D , approximately 9(c) Hence « - ^.(2 VoTd + p) (1 + ^ ) i . e . ^ , ( i + v j ^ ^ ^ i ^ ^jOj V^Td V (tj^/d)Eliminating I from equations9(c) and (10) gives
# - 2. A - A ^ ( U a ) d
where A = I (i + 1 ) ( n - ^ 1 ^ ^ ( l i b )
This solution yields two real and positive values of t ^ / d for each value of B, when v/V is l e s s than i.(V5 - 1), as shown in Fig. 16. Again it is suggested that the smaller of the two solutions is the correct one for this problem.
2
When v/V is snaall A can be neglected and hence
^L
- 2 A2.1 V
D . d • V
V
= 2d. — , as in section 14.
Finally, it may be remarked that the above analysis, for the case m > * , is applicable to the case of a very light depth of cut where the length of contact between wheel and work is very small.
16. Specific Energy
Fig. 17 shows the forces acting on a grinding wheel. The horizontal cutting force is F „ and the vertical normal force is F^,., The transverse force F _ opposes
H V I
the sense of the continuous cross t r a v e r s e . It may be neglected when the cross t r a v e r s e is zero or when the traverse is intermittent, as in reciprocating surface grinding,
In surface grinding the work done per unit time is
Fjj (V t v)
The volume of metal removed per unit time is
V. w,d
where w is the effective width of the wheel (Section 23).
The specific energy (u') is defined as the work done per unit volume of metal removed, thus
F „ ( V l : v)
U' = -T—
V, w,d
w.d V
The energy required per unit cross section area of metal removed is F „ ( V t v)
U = ^ w.v = u ' , d
For a given material the specific energy is found experimentally to be a function of the chip depth of c u t . ' ' • °' ' Fig, 18 shows a typical plot of u' against the maximum undeformed chip thickness t . The specific energy increases with decreasing chip thickness, as in metal cutting, up to the point A where, for dry surface grinding. it tends to remain constant. At A, in a typical case of grinding a hardened s t e e l ' ° '
22 -u' u' = 30. u 0.0092.u M c M c u' = u = 10.10^ in. -Ib/in^. c t^, = t = 40.10"^ in. M c
The following table compares the specific energy for grinding, fracture of a mild steel specimien and cutting with a single point tool.
fracture of M. S. u' = 10 i n , - l b / i n , 5
cutting = 5.10
a
grinding = 10.10
If u is defined as the specific energy required to cut a given material with a
15 rake single point tool and a feed rate (t) of 0,01 in, / r e v . , the following approximate relations have been suggested:''''
(13a)
(13b)
where nd[h0,8 .
The increase in specific energy with a decrease in undeformed chip thickness has been ascribed to the following e f f e c t s : ' ^ '
1. A size effect; the strength of a material increases with decreasing grain size when the cross section is of the same order of magnitude as the grain size.
2. The dulled edge of the cutting grit has an increasing relative effect on the cutting force per grit when the chip is small.
3. The total energy U is made up of two parts - a shear energy U and a friction energy U,, i . e .
U = U + U. s f
The shear energy decreases as the chip thickness is reduced. The friction energy remains sensibly constant and relatively independent of the chip thickness. Hence a decrease in chip thickness results in U, being a greater proportion of the total energy. The specific energy therefore tends to increase as the chip thickness is reduced,
F o r small thicknesses of chip the specific energy is sensibly constant. This appears to indicate a cutting process of a different nature. It is suggested that the removal of material may now take place by abrasion rather than by true cutting. The swarf appears as an ash rather than as chips.
In surface grinding a type 1 chip is normally formed. On the other hand it has been pointed out that, for internal cylinder grinding, the flat portion of the u' - t
(8)
curve is associated with a type III chip , Fig. 19, As the chip thickness is decreased further the specific energy tends to increase again. This portion of the curve is associated with a type v chip. The increase in energy is said to be due to the greater relative effect of chip-grit interface friction, due to the greater length of the chip^^).
17. Grit F o r c e s
The work done in a surface grinding process may be written
u ' . ( b , d ) . v per unit time
The number of chips produced per unit time is
V. k
= V . b . C
The work done per chip i s , therefore
u ' . b . d . V V . b . C
u' V _, =• c - V - ^
The mean force P on a grit is = work done/chip ° undeformed chip length
u ' . v d ' C.V ' I Also, since ^ ^ j
t = r ^
-M •vJ C . r • V * •vj C . r • V • (9) it follows that F = u ' . r . t ^-The maximum force per grit may be assumed to be twice the mean force, and is therefore
F^, = 2 . F M
= 2 . u ' . r . t f „ M
™,- -i-. XX. T^, force per grit The specific cutting p r e s s u r e , F ' = . .^—
F M b . t M 2 . u ' . r . t J . = M 4 . U '
24
For very fine grinding (t < t ) u' is independent of t . Thus 2 F - , =» (constant), t , , M M F o r thicker chips ( t , . > t ) M c constant u' = where 0.6 < n < 0.9 . Hence F = (constant), t
18, Cutting properties of a wheel
The cutting properties of a wheel are characterised
by:-the rate of metal removal the rate of wear
the surface finish produced the tendency to "load" the tendency to "burn" .
Loading of a wheel is the collection of metallic chips in the voids which are not dislodged when the grits clear the work. Burning of the work surfaces a r i s e s from excessive temperature r i s e at the work surface.
Each of the cutting properties are a function of:
(a) the inherent properties of the wheel, according to its grade, structure and so on.
(b) The apparent properties of the wheel as modified by the task set in the particular grinding operation,
The task set may be simply described in t e r m s of the load per grit and the volume of the mean chip relative to the void volume available.
19, Intrinsic Hardness
Consider unit volume of a wheel, taken near the surface. Let the volumes of grit, bond and voids be V , V, and V respectively. The following relation is satisfied g o p
V + V. + V = 1 g b p
If the grit volume is made up of n grits per unit of total volume, each of mean volume V then V = n.v , and
g g g
n . v + V^ + V = 1 g b p
The relative volumes of grit, bond and pores are not independent. For exajnple, an increase in bond volume, i . e . in strength of bond post, is balanced by a loss of porosity and/or grit volume. On the other hand it might be possible to decrease the number of grits n while increasing the grit size so as to leave V , and therefore
V, and V , unaltered, ^ b p
Suppose that for a particular grinding process it is possible to specify the length, thickness and volume of the mean chip. The load on a mean grit is then determined in principle. Then, for a given abrasive the tendency for the grit to fracture is some function of the mean volume per grit, v . The tendency for a bond post to
o
fracture is a function of V, , being greater as the bond post volume is decreased. The intrinsic hardness of a wheel is connected with its strength, i . e . the converse of its tendency to lose g r i t s . Thus when confronted with a standard task the strength might be increased by increasing v and V, . If v is increased relative to V the tendency to bond post fracture, rather than grit fracture, is Increased.
20, Wheel Loading
As already seen, an increase in v or V, will be accompanied by a decrease in the relative void fraction. This nnight be acceptable so long as the voids remaining a r e , on average, sufficient to accomnriodate the volume of the mean
grit chip. Moreover, a decrease in the space available for voids may be accompanied by a decrease in the volume of the mean void. There will be a limiting state in
which the void space is no longer able to cope with the volume of the mean chip, At this point V , or the volume of the mean void, v , is too small relative to the
P P chip volume. The chip tends to score and rub on the work, producing a defective surface and a tendency to grinding burn. The chip may become embedded in the pore and is not released when the cut of a grit is completed. This state of affairs will occur when the decrease in void volume is accompanied by an increase in bond post strength. On the other hand it may happen that the increased bond post strength is now insufficient for the imposed task. The overloaded bond posts tend to fracture and release the chips. Although chips may no longer be left embedded in the wheel there may still be a tendency for the wheel to score and burn.
The elimination of wheel loading niay be obtained
(a) at the expense of bond post volume and strength
(b) by reducing grit diameter, which is accompanied by a reduction in grit strength and an increase in the tendency towards grit fracture. This is not normally desirable, for grit fracture tends to produce two or more smaller grits without additional void space for the extra g r i t s .
(c) by reducing the number of grits per unit volume. In practice this will alter both the size of chip produced and the surface finish (C is a function of n).
The table below illustrates the probable effects on wheel characteristics of changes in inherent properties of the wheel. A positive sign refers to an increase in the quantity considered, a negative sign to a d e c r e a s e . The table assumes a given task, i . e . chip thickness and volume, and neglects the additional effect of changes in C on the task.
26 V^ V^ V- C bo'^d p°«t g - " loading g g p b fracture fracture - - . . . + . + + + - . . + - + + . + . - + + . + . + - , + + . + . . . + ~ . 4" • " 21. Relative Hardness
It may be shown that the relative hardness of a wheel is altered by the grinding conditions.
The mean force on a grit is
and u' u' - 30.u 0.0092.u F g . *2 t < t c t > t c 2
It follows that, for very snaall chips, the force per grit varies as t . F o r t g r e a t e r than the critical value t the force per grit varies as t ^ ; i . e . as t ' If n ^ 0.8
X
Since t . .
M
[c-^-^JF^]-it follows that t , , and F are both Increased, and the effective hardness is decreased,
1. decreasing work speed v,
2. Increasing wheel speed N, or wheel diameter D
3. increasing the number of active grits C ( i . e . by dressing technique, change of grain size or spacing)
4. decreasing the wheel depth of cut d
5. increasing the work diajneter D in external grinding, or decreasing the work diameter in internal grinding.
22. Wheel Wear
Fig. 20 illustrates the p r o g r e s s of total wheel wear, as measured by the total volume of wheel lost, plotted on a base of total volume of metal removed. After initial dressing the rate of wear of the fresh wheel is high (OA in Fig. 20). Over the normal operating range AB the rate of wear is sensibly constant. The grinding
ratio is the ratio
rate of naetal removal rate of wheel wear
X
y
and is measured on the straight portion of the wear curve.
It has been suggested that the nature of wheel wear may be by attrition, fracture or by chemical action^^^',
Attritions wear is the wearing away of the sharp cutting edges of the g r i t s . This is said to correspond to the flat portion of the wear curve'^^'. Eventually a flat surface is produced on the grit and the load on the grit tends to increase. The grit load may then be sufficient to rijpture either grit or bond post and
fracture wear is said to result. The wear rate then r i s e s sharply (curve BC, Fig. 20). If the bond posts or grits do not fracture the dull grits remain and the wheel tends to become glazed. It is suggested in the reference cited that if wear is purely
mechanical in nature, and if the forces on the grits are insufficient to cause fracture wear, then neither the wheel grade nor mechanical variables (wheel settings) should affect the grinding ratio. This has been borne out by experiments on plunge grinding'^^' On the other hand, where fracture wear does occur wheel wear is dependent on the mechanical variables (wheel and work speed and so on), wheel grade and the total metal renaoved.
Under certain conditions chemical wear m^ay occur. This a r i s e s from chemical reactions between grit and work. The tendency towards chemical wear is greatest at high grit face - chip interface t e m p e r a t u r e s . Apparently chemical wear is not normally important when grinding steel with an aluminium oxide wheel. Chemical reactions between grit and work are important when grinding titanium for example.
The foregoing is the conventional interpretation of the process of grinding wheel wear. On the other hand it has been shown'^^' that in surface grinding the wear at the periphery of the wheel is non uniform. When this is taken into account an alternative interpretation of the wear process is possible. This will be discussed in the following section.
23. Wear Profile in Surface Grinding (12)
It has been shown experimentally that, for surface grinding, the wear at the wheel periphery is non-uniform. Fig. 21 illustrates the characteristic profile obtained under normal cutting conditions. Initially, after dressing, the leading edge A of the wheel is square. Grits at this edge are relatively unsupported and weakly bonded to the wheel. When cutting s t a r t s the wheel crumbled away at the leading edge and the wear is comparatively rapid, as is found on the initial portion of a wear curve. Eventually the stable profile BCD is established, in which each grit has roughly the same support. F u r t h e r wear takes place at the active cutting face CD. The result is that the face CD r e t r e a t s uniformly a c r o s s the periphery towards the trailing edge E, That i s , the distance BC, =» X, i n c r e a s e s , leaving the shape of the profile unaltered. This regime of stable wear rate corresponds to the linear portion of the wear curve. Finally, when the trailing edge D of the profile reaches the face E of the wheel, this edge, being relatively unsupported,
28
-crumbles away. This regime corresponds with the final, relatively steep, portion of the wear curve.
In contrast with the wear theory advanced in the previous section normal wear on the linear portion of the wear curve takes place by both attrition and fracture on the active face C .
In Fig. 21 the cross feed per t r a v e r s e of the wheel is CC' = DD' = f. F o r surface grinding of flat surfaces the cross feed takes place intermittently at the end of each t r a v e r s e , before the wheel recommences its cut. The wheel profile shown in the figure was obtained under these conditions. In cylindrical grinding, on the other hand, the cross feed is continuous and f may be interpreted as the c r o s s feed per grit. That is
where s is the c r o s s feed rate in ft/min. , say.
In general, the c r o s s feed may be l e s s than, equal to, or greater than the effective width w of the wheel. Fig. 21, The condition f < w will be considered first, In this case the effective wheel depth of cut is less than the nominal geometric depth of cut AB, d. F o r the profile shown the maximum effective depth of cut is
C'C" = D'D"
= d e = f, tan a
and is independent of the nominal depth of cut.
The wear profile shown is fixed by the nominal depth of cut, the angle a of the wear land CD, and the distance X representing the current amount of total uniform wear. If it is assumed that the porosity of the wheel is sufficient to accommodate the chip formed then it may be shown that the angle of the wear land is a function of the task set the wheel. The following argument is due in part to the reference cited and is developed here on the basis of the theory given in the present report.
Consider therefore the effect of a change in cutting conditions. This change is experienced by the wheel as a change in the task per grit. The task per grit may be defined as the maximum grit force. F o r surface grinding this is a function of the specific energy and the maximum chip thickness. For most chip thicknesses the maximum force per grit varies as t ' , F o r cylindrical grinding, in which there is a continuous c r o s s feed, the effective hardness of the wheel is modified by
virtue of the lateral force associated with the cross feed. This is some function of the c r o s s feed and may be assumed to increase with the c r o s s feed. For the moment only surface grinding with intermittent c r o s s feed will be considered, The task set the mean grit then reduces to some function of the maximum chip thickness,
The wheel profile is self regulating and adjusts itself so that the task experienced by the mean grit tends to be equal to some critical value,
In effect, each grit tends to cut at a fixed force per grit F * (or a fixed maxlnaum chip thickness t*),
Evidently, for a given wheel and given machine settings (d, v, V, . . .
e t c . ) the only self adjustment allowed to the wheel is an alteration of the effective wheel depth of cut by a change of slope of the wear land. Since d = f, tan a, the effective depth of cut (and therefore chip thickness and task) can be increased by an increase in the angle of the wear land. Thus an increase in the severity of the cutting conditions can be compensated for by a change in a so a s to maintain the chip thickness unaltered.
M " N c . r V N D
where d has been substituted for d, the following changes in slope of the wear profile m a y b e predicted (a positive sign denotes an increase, a negative sign a decrease in the magnitude of the quantity considered).
d D V V C a +
The effect of a reduction in a for a constant nominal depth of cut is to produce a longer, flatter,wear profile CD.
Since the effective depth of cut, and therefore the chip thickness, is independent of the nominal depth of cut, changes in d leave the land angle unaltered.
In general the flatter the wear profile the greater will be both the wear rate and the rate of metal removal, for the effective cutting width of the wheel is increased.
F r o m the formula for d it is seen that an Increase in the intermittent c r o s s Q
feed requires a reduction in the angle a in order to maintain the same effective depth of cut. This is true only so long as the feed is l e s s than the effective width of the wheel.
If, on the other hand, the intermittent c r o s s feed is greater than the width w of the profile (f > w) it follows that the maximum effective wheel depth of cut is equal to the nominal depth of cut. Fig. 22. The self regulating action of the wheel is lost. If, however, the grits are still cutting within their capacity the profile
30
-Changes in cutting conditions which increase the task beyond some critical value cannot be accommodated by a wheel having this latter profile, for the effective depth of cut is now fixed. It may be expected that the wheel profile would crumble in o r d e r to establish another profile such that f is again less than w and the self-regulating characteristics of the wheel are regained.
If the c r o s s feed is continuous, as in cylindrical grinding, f is now the c r o s s feed per grit. The foregoing arguments still hold, except that the critical value of the task which is accepted by the mean grit is lower due to the t r a n s v e r s e force and the land angle is correspondingly smaller. An increase in continuous c r o s s feed rate will increase the lateral force on a grit and hence the wear land angle will be reduced still further to maintain the same task.
It may be concluded that a fundamental parameter is the working value of the critical task per grit. This is a function of the inherent properties of the wheel and is independent of the machine settings, (except that cross feed in continuous c r o s s t r a v e r s e may affect this value). For surface grinding the critical task is measured by a critical chip thickness given by
+ r i V
fd"
where d = f. tan a (f < w) e
= d (f > w)
In peripheral side wheel grinding it is possible that non uniform wheel wear occurs at the leading edge of the wheel. However it has not been found possible to predict a simple wear profile able to provide self regulation of the effective depth of cut. Similarly, there is no theoretical reason for a characteristic wear profile to occur in gear grinding, for example. However these conclusions have not been tested experimentally.
Finally, when calculating the average force on a grit and the rate of metal removal, the t e r m effective wheel width (w') has been used. For surface grinding this is defined as follows:
The rate of metal renaoval per unit (R) = w ' . d . v
But R = (area CC'DD').v (Fig. 21) = f . d . v
i . e . R = w ' . d . v = f . d . v
w' = f
= feed per t r a v e r s e
